, ⑤
: 1-
(closed form) -(n 1) +
(x) (A(x)B(11) +...
S
Geometric Sum functions Convolution Robn abn-1 dibn
Geneating = +
:et
1 x
Pick nelemt with B(x)K flavon e
en (n h 1)
-
2 flavors + -
!)
=
(n+ )
=
n
Perturbation method S -S
↳ 1 + ,
·
: n
N
Sumofpowers : (2n + 1) (n + 1) n
6
n ! Ti
Only one
possibility 3 Tower of Hanoï
Closed forms
i=
x
T(x) =
Approximation +
f(1) f(i) f(n + Select 6 apples +
1+
*
+x
*
+... xbn =
1
1 x6
(1 (x)(1 x)
- -
[c
k= 1
= c + c +... + c = nC
( f(x)dx nbiseven 1
nbogsteps +n (xn)T(x) 2
-
I = - : = =
-
1 U
,
x2
x =
1
k
-
Infinite Geometric series If : (1 = =
1 + 2 +... + n
=
i 1 x 1
2 decks of
-
1+ 1/2 + 14 +... (2) =
i) nb multiple of 5 52 cards -
104 cards
x5
=1
0999
2 +... + n
of
1 mb distinct permutations
-
+ =
-52
n(n + 1)(2n + 1)
- 104n1
2 2
Hnzen(n) use approx
.
types of bananas n= = nz =... =
152 =
6
/Sex xSn) ... = (Sel ..... ISn) = Si s Atmost4s 1-x5 Book =
4 ! /2 !
10 !
= 13 + 2 ...
/AnUAzU
1 x
Bookkeeper /2 !. 21 3! n2(n 1)2
-
(An) (An)
+
...
= + (Az) +... + (An I =
.
=
! 4
(2)
n
=
possibilities S At most one -
1 +x Binomial
coeff of c in (1 +x)
& x
(n-k) ! k!
odd nb x+ 13 + x5 theorem
S - -
=
- 1/ *
(mm) (1)1" x5 (5)x5
x2
Multinomial Coeff :
Il At least 2 - -2 1 -
( + x)
= ) 111 .
- 4
.
xk = =
O 1 x
mi
-
Binomial Theorem : (a+ b)" =) an x3 +... (1 +x)
1
-
.
1 -
x (2
+
-
= (1 + xz +... + 71 = 100
(1) (1 -1) (19)
2 -
1 -
2x + 3x2 4x3 (1 +x)
-
=
= =
2
Pigeonhole principle If IAK K /BI then for 2x + 3x 4x
-
(1 x)
everyelemt
: 1 + + =
-
.
, ..
Total function f : A -B there exist k + 1 + 1 +x +x + x +x =
x5 -
1/x -
1
Integrals
3
of t that are mapped by f to the same elemt of B
. 1 + 3x + 3x + x ? = (1 x) +
1
Function
kEIR
f primitive F
(AuBI kx + C
-
Inclusion Exclusion (A) 1BI (AMBI 1 + ax + a2x + a3 = (1 ax)
-
-
: = + - -
xn+ 1
153) ISenSal-IS11Sz/ x" nEIN
-2
ISeUSzUSzl = (Sel + 152) + -
1 + 3x + 9x2 + 27x3. = (1 3x) -
,
n+ 1 + C
(S2nSz) /S11S2nS3) 9x2 27x 3. (1
1
1/Fr 25 + C
-
-
+ 1 -
3x + -
= + 3x)
Prof(n) (n'k) 1/xn 1/(n 1)xn 1
2
combinatorial 3x+ 4x3
- -
1 + 2x + (1 x)
-
-
-
= =
...
(n) (m) ( *i1) u'(x)u(x) un 1(x)/n + 1
2
Pascal's 2x2 + 3x3 +
-
triangle -dentity
T = = + x + ... = x(1 x) -
VANDERMONDE's identity = ) (ru) (min) =
x3 + 2x" + 3x5 = x3/(1 x)2 -
u'(x)/Ju(x) aut
Linear Recurrences : Fibonacci x3 + x" + x5 ... = x3(1 x) -
u'()/u(x) en (u(x))
F() =
fo f(x f2x2
+ + +... +
frx" +... 1 + x + x3 x + x5
+
.
=
(1 -
x)- x-
2 -
2x + 2x2 2x -
= 2/(1 + x) x
...
F(x) = (1 3x)(1 x))
- -
1 x x2
-
-
x
-
(1-3x)(1 x)(1 + x) -
: 1-
(closed form) -(n 1) +
(x) (A(x)B(11) +...
S
Geometric Sum functions Convolution Robn abn-1 dibn
Geneating = +
:et
1 x
Pick nelemt with B(x)K flavon e
en (n h 1)
-
2 flavors + -
!)
=
(n+ )
=
n
Perturbation method S -S
↳ 1 + ,
·
: n
N
Sumofpowers : (2n + 1) (n + 1) n
6
n ! Ti
Only one
possibility 3 Tower of Hanoï
Closed forms
i=
x
T(x) =
Approximation +
f(1) f(i) f(n + Select 6 apples +
1+
*
+x
*
+... xbn =
1
1 x6
(1 (x)(1 x)
- -
[c
k= 1
= c + c +... + c = nC
( f(x)dx nbiseven 1
nbogsteps +n (xn)T(x) 2
-
I = - : = =
-
1 U
,
x2
x =
1
k
-
Infinite Geometric series If : (1 = =
1 + 2 +... + n
=
i 1 x 1
2 decks of
-
1+ 1/2 + 14 +... (2) =
i) nb multiple of 5 52 cards -
104 cards
x5
=1
0999
2 +... + n
of
1 mb distinct permutations
-
+ =
-52
n(n + 1)(2n + 1)
- 104n1
2 2
Hnzen(n) use approx
.
types of bananas n= = nz =... =
152 =
6
/Sex xSn) ... = (Sel ..... ISn) = Si s Atmost4s 1-x5 Book =
4 ! /2 !
10 !
= 13 + 2 ...
/AnUAzU
1 x
Bookkeeper /2 !. 21 3! n2(n 1)2
-
(An) (An)
+
...
= + (Az) +... + (An I =
.
=
! 4
(2)
n
=
possibilities S At most one -
1 +x Binomial
coeff of c in (1 +x)
& x
(n-k) ! k!
odd nb x+ 13 + x5 theorem
S - -
=
- 1/ *
(mm) (1)1" x5 (5)x5
x2
Multinomial Coeff :
Il At least 2 - -2 1 -
( + x)
= ) 111 .
- 4
.
xk = =
O 1 x
mi
-
Binomial Theorem : (a+ b)" =) an x3 +... (1 +x)
1
-
.
1 -
x (2
+
-
= (1 + xz +... + 71 = 100
(1) (1 -1) (19)
2 -
1 -
2x + 3x2 4x3 (1 +x)
-
=
= =
2
Pigeonhole principle If IAK K /BI then for 2x + 3x 4x
-
(1 x)
everyelemt
: 1 + + =
-
.
, ..
Total function f : A -B there exist k + 1 + 1 +x +x + x +x =
x5 -
1/x -
1
Integrals
3
of t that are mapped by f to the same elemt of B
. 1 + 3x + 3x + x ? = (1 x) +
1
Function
kEIR
f primitive F
(AuBI kx + C
-
Inclusion Exclusion (A) 1BI (AMBI 1 + ax + a2x + a3 = (1 ax)
-
-
: = + - -
xn+ 1
153) ISenSal-IS11Sz/ x" nEIN
-2
ISeUSzUSzl = (Sel + 152) + -
1 + 3x + 9x2 + 27x3. = (1 3x) -
,
n+ 1 + C
(S2nSz) /S11S2nS3) 9x2 27x 3. (1
1
1/Fr 25 + C
-
-
+ 1 -
3x + -
= + 3x)
Prof(n) (n'k) 1/xn 1/(n 1)xn 1
2
combinatorial 3x+ 4x3
- -
1 + 2x + (1 x)
-
-
-
= =
...
(n) (m) ( *i1) u'(x)u(x) un 1(x)/n + 1
2
Pascal's 2x2 + 3x3 +
-
triangle -dentity
T = = + x + ... = x(1 x) -
VANDERMONDE's identity = ) (ru) (min) =
x3 + 2x" + 3x5 = x3/(1 x)2 -
u'(x)/Ju(x) aut
Linear Recurrences : Fibonacci x3 + x" + x5 ... = x3(1 x) -
u'()/u(x) en (u(x))
F() =
fo f(x f2x2
+ + +... +
frx" +... 1 + x + x3 x + x5
+
.
=
(1 -
x)- x-
2 -
2x + 2x2 2x -
= 2/(1 + x) x
...
F(x) = (1 3x)(1 x))
- -
1 x x2
-
-
x
-
(1-3x)(1 x)(1 + x) -
